linear.approx {boot} | R Documentation

## Linear Approximation of Bootstrap Replicates

### Description

This function takes a bootstrap object and for each bootstrap replicate it calculates the linear approximation to the statistic of interest for that bootstrap sample.

### Usage

    linear.approx(boot.out, L=NULL, index=1, type=NULL, t0=NULL,
                  t=NULL, ...)

### Arguments

`boot.out` | An object of class `"boot"` representing a nonparametric bootstrap. It will usually be created by the function `boot`.
`L` | A vector containing the empirical influence values for the statistic of interest. If it is not supplied then `L` is calculated through a call to `empinf`.
`index` | The index of the variable of interest within the output of `boot.out$statistic`.
`type` | This gives the type of empirical influence values to be calculated. It is not used if `L` is supplied. The possible types of empirical influence values are described in the helpfile for `empinf`.
`t0` | The observed value of the statistic of interest. The input value is used only if one of `t` or `L` is also supplied. The default value is `boot.out$t0[index]`. If `t0` is supplied but neither `t` nor `L` are supplied then `t0` is set to `boot.out$t0[index]` and a warning is generated.
`t` | A vector of bootstrap replicates of the statistic of interest. If `t0` is missing then `t` is not used, otherwise it is used to calculate the empirical influence values (if they are not supplied in `L`).
`...` | Any extra arguments required by `boot.out$statistic`. These are needed if `L` is not supplied as they are used by `empinf` to calculate empirical influence values.

### Details

The linear approximation to a bootstrap replicate with frequency vector `f` is given by `t0 + sum(L * f)/n` in the one sample with an easy extension to the stratified case. The frequencies are found by calling `boot.array`.

### Value

A vector of length `boot.out$R` with the linear approximations to the statistic of interest for each of the bootstrap samples.

### References

Davison, A.C. and Hinkley, D.V. (1997) Bootstrap Methods and Their Application. Cambridge University Press.

### See Also

`boot`, `empinf`, `control`

### Examples

    # Using the city data let us look at the linear approximation to the
    # ratio statistic and its logarithm. We compare these with the
    # corresponding plots for the bigcity data

    data(city)
    data(bigcity)
    ratio <- function(d, w)
         sum(d$x * w)/sum(d$u * w)
    city.boot <- boot(city, ratio, R=499, stype="w")
    bigcity.boot <- boot(bigcity, ratio, R=499, stype="w")
    par(pty="s")
    par(mfrow=c(2,2))

    # The first plot is for the city data ratio statistic.
    city.lin1 <- linear.approx(city.boot)
    lim <- range(c(city.boot$t,city.lin1))
    plot(city.boot$t, city.lin1, xlim=lim,ylim=lim,
         main="Ratio; n=10", xlab="t*", ylab="tL*")
    abline(0,1)

    # Now for the log of the ratio statistic for the city data.
    city.lin2 <- linear.approx(city.boot,t0=log(city.boot$t0),
                               t=log(city.boot$t))
    lim <- range(c(log(city.boot$t),city.lin2))
    plot(log(city.boot$t), city.lin2, xlim=lim, ylim=lim,
         main="Log(Ratio); n=10", xlab="t*", ylab="tL*")
    abline(0,1)

    # The ratio statistic for the bigcity data.
    bigcity.lin1 <- linear.approx(bigcity.boot)
    lim <- range(c(bigcity.boot$t,bigcity.lin1))
    plot(bigcity.lin1,bigcity.boot$t, xlim=lim,ylim=lim,
         main="Ratio; n=49", xlab="t*", ylab="tL*")
    abline(0,1)

    # Finally the log of the ratio statistic for the bigcity data.
    bigcity.lin2 <- linear.approx(bigcity.boot,t0=log(bigcity.boot$t0),
                                  t=log(bigcity.boot$t))
    lim <- range(c(log(bigcity.boot$t),bigcity.lin2))
    plot(bigcity.lin2,log(bigcity.boot$t), xlim=lim,ylim=lim,
         main="Log(Ratio); n=49", xlab="t*", ylab="tL*")
    abline(0,1)

    par(mfrow=c(1,1))

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